A graph is -colorable if its vertices can be partitioned into subsets and such that every vertex in has degree at most for each . We prove that every graph with maximum average degree at most is -colorable. In particular, it follows that every planar graph with girth at least is -colorable. On the other hand, we construct graphs with maximum average degree arbitrarily close to (from above) that are not -colorable.
In fact, we establish the best possible sufficient condition for the -colorability of a graph in terms of the minimum, , of over all subsets of . Namely, every graph with is -colorable. On the other hand, we construct infinitely many non--colorable graphs with . This solves a related conjecture of Kurek and Ruciński from 1994.